How Do Large Language Monkeys Get Their Power (Laws)?

TL;DR

This paper investigates why the success rate of multimodal language models scales as a power law with attempts, revealing that heavy-tailed success probability distributions explain the phenomenon and improve scaling predictions.

Contribution

It uncovers the heavy-tailed distribution of success probabilities as the key to polynomial scaling, providing a new perspective and method for predicting model performance.

Findings

Heavy-tailed success probability distribution explains polynomial scaling.

Empirical confirmation of exponential decay in failure rate per attempt.

Method for more accurate power law exponent forecasting with less compute.

Abstract

Recent research across mathematical problem solving, proof assistant programming and multimodal jailbreaking documents a striking finding: when (multimodal) language model tackle a suite of tasks with multiple attempts per task -- succeeding if any attempt is correct -- then the negative log of the average success rate scales a power law in the number of attempts. In this work, we identify an apparent puzzle: a simple mathematical calculation predicts that on each problem, the failure rate should fall exponentially with the number of attempts. We confirm this prediction empirically, raising a question: from where does aggregate polynomial scaling emerge? We then answer this question by demonstrating per-problem exponential scaling can be made consistent with aggregate polynomial scaling if the distribution of single-attempt success probabilities is heavy tailed such that a small fraction of tasks with extremely low success probabilities collectively warp the aggregate success trend into a power law - even as each problem scales exponentially on its own. We further demonstrate that this distributional perspective explains previously observed deviations from power law scaling, and provides a simple method for forecasting the power law exponent with an order of magnitude lower relative error, or equivalently, ${\sim}2-4$ orders of magnitude less inference compute. Overall, our work contributes to a better understanding of how neural language model performance improves with scaling inference compute and the development of scaling-predictable evaluations of (multimodal) language models.